Theorem lspsolv 21248

Description: If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lspsolv.v	⊢ 𝑉 = (Base‘𝑊)
lspsolv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspsolv.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspsolv	⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Proof of Theorem lspsolv

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsolv.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		lspsolv.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspsolv.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		eqid 2763	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2763	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
7		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
8		eqid 2763	. . 3 ⊢ {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)} = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)}
9		lveclmod 21208	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	9	adantr 485	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑊 ∈ LMod)
11		simpr1 1213	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝐴 ⊆ 𝑉)
12		simpr2 1214	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ 𝑉)
13		simpr3 1215	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
14	13	eldifad 3918	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14	lspsolvlem 21247	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
16	4	lvecdrng 21207	. . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
17	16	ad2antrr 738	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (Scalar‘𝑊) ∈ DivRing)
18		simprl 782	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
19	10	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑊 ∈ LMod)
20	12	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ 𝑉)
21		eqid 2763	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
22		eqid 2763	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
23	1, 4, 7, 21, 22	lmod0vs 20997	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
24	19, 20, 23	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
25	24	oveq2d 7428	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)(0g‘𝑊)))
26	11	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ 𝑉)
27	20	snssd 4753	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑌} ⊆ 𝑉)
28	26, 27	unssd 4146	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑌}) ⊆ 𝑉)
29	1, 3	lspssv 21085	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑌}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
30	19, 28, 29	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
31	30	ssdifssd 4102	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)) ⊆ 𝑉)
32	13	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
33	31, 32	sseldd 3939	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ 𝑉)
34	1, 6, 22	lmod0vrid 20995	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
35	19, 33, 34	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
36	25, 35	eqtrd 2798	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = 𝑋)
37	36, 32	eqeltrd 2863	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
38	37	eldifbd 3919	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
39		simprr 784	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
40		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
41	40	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)))
42	41	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → ((𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) ↔ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
43	39, 42	syl5ibcom 248	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
44	43	necon3bd 2972	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) → 𝑟 ≠ (0g‘(Scalar‘𝑊))))
45	38, 44	mpd 16	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ≠ (0g‘(Scalar‘𝑊)))
46		eqid 2763	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
47		eqid 2763	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
48		eqid 2763	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
49	5, 21, 46, 47, 48	drnginvrl 20842	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
50	17, 18, 45, 49	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
51	50	oveq1d 7427	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
52	5, 21, 48	drnginvrcl 20839	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
53	17, 18, 45, 52	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
54	1, 4, 7, 5, 46	lmodvsass 20989	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
55	19, 53, 18, 20, 54	syl13anc 1399	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
56	1, 4, 7, 47	lmodvs1 20992	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
57	19, 20, 56	syl2anc 595	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
58	51, 55, 57	3eqtr3d 2806	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
59	33	snssd 4753	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑋} ⊆ 𝑉)
60	26, 59	unssd 4146	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ 𝑉)
61	1, 2, 3	lspcl 21078	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
62	19, 60, 61	syl2anc 595	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
63	1, 4, 7, 5	lmodvscl 20980	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
64	19, 18, 20, 63	syl3anc 1398	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
65		eqid 2763	. . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊)
66	1, 6, 65	lmodvpncan 21017	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
67	19, 64, 33, 66	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
68	1, 6	lmodcom 21010	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
69	19, 64, 33, 68	syl3anc 1398	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
70		ssun1 4132	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑋})
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ (𝐴 ∪ {𝑋}))
72	1, 3	lspss 21086	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑋})) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
73	19, 60, 71, 72	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
74	73, 39	sseldd 3939	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
75	69, 74	eqeltrd 2863	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
76	1, 3	lspssid 21087	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
77	19, 60, 76	syl2anc 595	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
78		snidg 4627	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
79		elun2 4137	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐴 ∪ {𝑋}))
80	33, 78, 79	3syl 19	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝐴 ∪ {𝑋}))
81	77, 80	sseldd 3939	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
82	65, 2	lssvsubcl 21046	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})) ∧ 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
83	19, 62, 75, 81, 82	syl22anc 851	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
84	67, 83	eqeltrrd 2864	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
85	4, 7, 5, 2	lssvscl 21057	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
86	19, 62, 53, 84, 85	syl22anc 851	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
87	58, 86	eqeltrrd 2864	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
88	15, 87	rexlimddv 3172	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  {crab 3416   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  {csn 4590  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  ._rcmulr 17312  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  -_gcsg 19003  1_rcur 20264  inv_rcinvr 20470  DivRingcdr 20814  LModclmod 20962  LSubSpclss 21033  LSpanclspn 21073  LVecclvec 21204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-tpos 8223  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-minusg 19005  df-sbg 19006  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-oppr 20420  df-dvdsr 20440  df-unit 20441  df-invr 20471  df-drng 20816  df-lmod 20964  df-lss 21034  df-lsp 21074  df-lvec 21205

This theorem is referenced by:  lssacsex  21249  lspsnat  21250  lsppratlem1  21252  lsppratlem3  21254  lsppratlem4  21255  lbsextlem4  21266  lindsadd  38245  lindsenlbs  38247